# Full text of "Physics 101, Black Holes and Gravitational Fields Assignment"

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```Physics o P Black Holes

Patrick Bruskiewich

June, 2002

t

970

CHAPTER

unitization

Figure 1

How the geometry of space is
warped by a spherical (nonro-
distance from any point to the
center of the black hole can be
measured in two ways. One
way is to measure ihe circum-
ference of a circle passing
through the point and cen-
tered on the hole, and divide
by 2tt. A second way is to
measure the distance from the
point to the center along a
radial line on the curved sur-
face. These two measures will
be nearly equal far from the
hole where space is "flat,"
but highly discrepant near
the black-hole horizon where
the funnel narrows to a ver-
tical tube.

Black Holes

Alan P. Lightman
Cornell University

Imagine a region of space where attractive gravitational force is so intense that
light rays venturing too near can be bent into circular orbit -a region from
which no matter, radiation, or communication of any kind can ever escape.
Such is a black hole, one of the most exciting creatures of theoretical physics
and possibly the most bizarre object of space.

Although they were implicitly predicted by Einstein's 1915 theory of grav-
ity, general relativity, black holes were first theoretically "discovered" by Op-
penheimer and Snyder in 1939. Because of their highly nonintuitive proper-
ties, however, black holes were not taken seriously by most physicists and
astronomers until the mid-1960s. We are now on the verge of confirming the
discovery in space of the first black hole.

The concept of a black hole stretches our ideas of time and space to their
limits. The surface of a black hole, called its horizon, is a closed boundary
within which the escape velocity exceeds the velocity of light. The prediction
of such a surface for sufficiently compact bodies can be made just on the basis
of Newton's theory of gravity together with special relativity: the escape veloc-
ity of a particle launched from the surface of a spherical mass M of radius R is
v e = V2GM/R (see Chapter 9). When MIR satisfies 2GM/R > c 2 , v e exceeds the
speed of light and no particle or photon may escape, as required by special rel-
ativity. As a remarkable result, the interior of a black hole is causally discon-
nected from the rest of the universe; no physical process occurring inside the
horizon can communicate its existence or effects to the outside. For a spherical
black hole of mass M, the horizon is a sphere of circumference equal to 2ir
times the Schwarzschild radius of the hole Re, where Re = 2GM/c 2 (exact nu-
merical coincidence of this radius with the newtonian analog above is ac-
cidental). A black hole with a mass equal to that of our sun would have a

According to general relativity, time and space are warped by the gravita-
tional field of massive bodies, with the warping most severe near a biack hole.
Gravity affects all physical systems in a universal way so that all clocks
(whether transitions of an ammonia molecule or heartbeats of a human being)
and all rulers would indicate that time is slowed down and space stretched out
near a black hole (see Figure 1). Alternatively, one can describe the effects of
the hole's gravitational field on a local measurement of time and space in-'
tervals as an acceleration of the local Lorentz frame (in which special relativity
is valid) with respect to other local Lorentz frames at different locations.

Black holes are formed when massive stars undergo total gravitational
collapse. While emitting heat and light into space, stars balance themselves
against their own gravity with the outward thermal pressure force generated
by heat from nuclear reactions in their deep interiors. But every star must die.
When its nuclear fuel is exhausted, a star will contract. If its mass is Jess fen
about three times the mass of our sun, the shrinking star will stabilize at a
smaller size when the inward pull of gravity cannot force the star's constituent
particles any closer together. Such a star will spend eternity as a white dwarf or
neutron star. But if the star's mass exceeds about 3 solar masses, theory in-
dicates that no amount of outward pressure force can stave off the overwhelm-
ing crush of gravity and the star will plunge in upon itself, disappearing for-
ever from sight and giving birth to a black hole.

Recent mathematical proofs, using Einstein's general relativity theory, dem-
onstrate that a black hole is one of the simplest objects of nature and can be
completely characterized by only three quantities: its mass, angular momen-
tum, and net electrical charge. Other than these three quantities, all informa-
tion of the progenitor star, e.g., whether it was composed of particles or
antiparticles, whether it was pancake-shaped or spherical, is lost via gravita-

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tonal and electromagnetic waves shortly after formation of the hole. Rotating

sur o, JTk C f h °u S (n ° nrotat!n g hol « ^ "lied SchwarzschM holes),

surround themselves with a reg,on called the ergasphere in which space and

■me are so strongly twisted that all particles, photons, and even local Lorentz

frames are forced to rotate about the hole. Lorentz

ppn A ' fTu 6iStanCe ! fr ° m a b ' aCk h ° !e ' ' tS 8 ravl ^tional field behaves as if
generated by an ordinary star of mass M, obeying the newtonian inverse-
square law. Close to a black hole, the gravitational field is far stronger than

mas" blar k P h1 1Cted ^ ^^^ theorv - A ™" being sucked into a solar-
mass black hole would be npped to shreds by the differential force of gravitv
across his body long before he reached the horizon

Once the hole's horizon has formed we cannot receive any information

lap" r dial dT^ ate °u ^ ?*«*"* "" ^ C ^°™ ° f ^ col

ume andTnr 6 > A ff' ^^ that Ae Star 1S Crushed *> «"> vol-
ume and mfmite dens.ty at the center of the black hole, forming there a point

of infinite gravitational force called. a singularity. For a collapsing star of a few

olar masses, the final stellar death throes would be over in a few hundred

la°ss^ t 3 f C ° nd ( f S meaSUr6d l0Cally) - Quantum effects ' omi «ed in the
classical theory of general relativity, could possibly halt the stellar collapse at

onsH T 8 7 t K " Sity P ~ C '' hGi ~ 5 X 10 " S m/Cm: ' Where * > s Pick's

prevent thTfc ^"Tu, ?*? ," ° f sin 8 uIa » tle ^ but such effects could not
prevent the formation of black holes.

, nn C ";, ent th J m ! eS u ° f Stellar evo,utl0n suggest that there could be as many as
100 million black holes in our galaxy. The search for these black holes is not
easy We could never detect a tiny black speck a few miles across against the
mght sky. Instead we must search for signs of black holes interacting with
their astrophysical neighborhoods. Most vulnerable to discovery would be a
black hole orbiting a normal star in a binary system. Using Kepler's laws, anal-
ysis of the magnitude and period of the doppler shift of the visible normal star
a lows us to compute whether the unseen companion is massive enough to be a
black hole In addition we might observe intense,, flickering x-rays produced
when gas from the normal star is sucked toward the black hole "and heated to
a billion degrees in its spiraling path to destruction in the hole. Such are the

o a nTk , k bmary X " aV S ° UrCe CygnuS X- 1 ' an e * ceil <^ «mdidate
for a black hole about 8000 light-years from earth in the constellation Cvgnus

Black holes are a fundamental phenomenon of nature. Space may be littered
with black holes tempting us with their secrets of time and space behind a
cloak of impenetrable darkness, a challenge and a prize for the persistence of
astronomers and physicists.

Name:

Black Hole Test

Per

Part A: True/False

For each of the following, circle the T if the statement is true. If the statement is false,
circle the F, and rewrite as a true statement on the line beneath.

T F 1. Light can escape from a black hole.

T F 2. Black holes were theoretically discovered by Oppenheimer

and Snyder in 1939.

T F 3.

The theory used to predict black holes is Einstein's Theory of
General Relativity.

T F 4. Clocks go faster near a black hole.

T F 5. Distances are shortened near a black hole.

T F 6. A black hole can be characterized by only three quantities: mass,

angular momentum and net electrical charge.

T F 7. By looking at a black hole you can tell how it was formed.

T F 8.

A black hole the mass of the sun would have a Schwarzschild

Part B : Matching

In the space to the left, write the letter of the description that best fits each term.

Term :
.9. Schwarzschild hole

.10. A Black Hole

.11. Ergosphere

.12. Kerr hole
.13. Horizon

.14. Light
.15. Singularity
.16. Space

Description :

A. an open boundary within which the escape
velocity exceeds the velocity of light.

B. venturing too near a black hole can be bent

into circular orbit.

C. point of infinite density at the centre of
a black hole

D. may be littered with black holes.

E. are formed when massive stars undergo
total gravitational collapse.

F. region around a stationary black hole.

G. a non-rotating black hole.

H. a rotating black hole.

I. a closed boundary within which the escape

velocity exceeds the velocity of light.

J. region around a spinning black hole.

Part C : Multiple Choice

In each of the following questions, put your answer in the space provided.

17. The theory that implicitly predicts the existence of black holes is? 17,

A. the special theory of relativity

B. the general theory of relativity

C. the theory of singularities

D. the theory of stellar evolution

18. Stars balance themselves against their collapse?

18

A. by gravitational force C.

B. by spinning very fast D.

by thermal pressure
by nuclear forces

19. A star will collapse into a black hole when

19.

A. its mass is less than three solar masses

B. it is spinning at three times that of the sun

C. its mass is greater than three solar masses

D. it starts to become a white dwarf or neutron star

20.

Once the black hole's horizon has been formed

20.

A. we cannot receive any information from within the black hole

B. its mass is crushed to zero volume at the centre

C. its mass is crushed to infinite density at the centre

D. all of the above

21.

To detect a black hole

21.

A. we can use a telescope to see then directly

B. we cannot see them with a telescope

C. we need to watch for the Doppler shift of a black hole

D. we need to watch for the Doppler shift of a companion star

22.

The interior of a black hole

22.

A. is causally connected with the outside world

B. is causually disconnected with the outside world

C. is visible to a distant observer using an optical telescope

D. is visible to a distant observer using a radio telescope

23. At great distance from a black hole

23.

A. space is stretched or warped

B. time slows down

C. space behaves as if generated by a normal star

D. we don't see or feel the black hole

24. A man falling into a black hole

24.

E. would not feel a thing

F. would find his watch going slower

G. would be ripped to shreds

H. would feel like he was be compressed

25. Using the principle of conservation of energy, prove that the radius of a
black hole of mass M is the Schwarzschild Radius Rs = 2 GM/c 2 . (4 marks)

26. Calculate the Schwarzschild Radius of the earth in metres. (4 marks)

mass of the earth = 5.98 x 10 24 kg

c = 3x10 8 m/s

G = 6.67x1CT 11 Nm 2 /kg 2

27. Describe how you might use a telescope to find a black hole

(100 words or less) (4 marks)

.

28. In a gravitational field the frequency of light changes as light climbs out of the
gravitational well. Starting with the expression for the change in gravitational
energy U = mgz where z is the height above the star and using the Planck
expression E = h v, where v is the frequency of the light find an expression for
the change in frequency of the emitted light. (6 marks)

29. Estimate the fractional change in frequency for light that climbs 10 metres
from the surface of the earth. (4 marks)

30. Given the expression for the fractional change in frequency for light that
climbs out of a gravitational well, speculate what happens to the light from a star
that begins to collapse into a black hole. ( 100 words or less) (4 marks)

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